(a) Determine the critical value for a right-tailed test of...

(a) Determine the critical value for a right-tailed test of a population mean at the $\alpha=0.01$ level of significance with 15 degrees of freedom. (b) Determine the critical value for a left-tailed test of a population mean at the $\alpha=0.05$ level of significance based on a sample size of $n=20$ (c) Determine the critical values for a two-tailed test of a population mean at the $\alpha=0.05$ level of significance based on a sample size of $n=13$

(a) Determine the critical value for a right-tailed test of a population mean at the $\alpha=0.01$ level of significance with 15 degrees of freedom.
(b) Determine the critical value for a left-tailed test of a population mean at the $\alpha=0.05$ level of significance based on a sample size of $n=20$
(c) Determine the critical values for a two-tailed test of a population mean at the $\alpha=0.05$ level of significance based on a sample size of $n=13$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, and then using a test statistic to determine whether to reject the null hypothesis based on a predetermined significance level.

t-Distribution

The t-distribution is a probability distribution used in statistical inference when the sample size is small and/or the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the increased uncertainty associated with smaller samples.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis without breaking any constraints. In the context of the t-distribution, it typically equals the sample size minus one, and it affects the shape of the distribution; fewer degrees of freedom result in a distribution with fatter tails.

Significance Level (?)

The significance level, denoted by ?, is the probability threshold set by the researcher for rejecting the null hypothesis. It represents the risk of committing a Type I error, i.e., incorrectly rejecting the true null hypothesis. Common values are 0.01 and 0.05.

Critical Value

The critical value is the threshold value that the test statistic must exceed in order to reject the null hypothesis in a hypothesis test. It is determined from the t-distribution (or another relevant distribution) based on the degrees of freedom, significance level, and whether the test is one-tailed or two-tailed.

One-Tailed and Two-Tailed Tests

One-tailed tests assess the probability of a test statistic being extreme in one direction (either right or left), while two-tailed tests evaluate extremity in both directions. The tail(s) determine how the significance level is split and, consequently, the positioning of the critical value(s) on the t-distribution.

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